2.019 Design of Ocean Systems

Lecture 5

Seakeeping (I)

February 18, 2011

Six-Degree-of-Freedom Motion of a Floating Body in Waves

β

Waves Propagating

yaw

heav

e sway

pitch

roll

z

Uy

x

surge

Image b

Translation in x: surge ζ1(t)Translation in y: sway ζ2(t);Translation in z: heave ζ3(t);Rotation with x: roll ζ4(t);Rotation with y: pitch ζ5(t);Rotation with z: yaw ζ6(t);

y MIT OpenCourseWare.

Local motions

Slamming

Water on deck

Effect of breaking waves

AccelerationsWave bending moments and shear forces

Liquid sloshing in Tanks

Examples of Seakeeping and Wave Load Problems for Ships andOffshore Structure

Image by MIT OpenCourseWare.

Concerns of Seakeeping in FPSO Design

• Increasing maximum loads (due to dynamic pressure)• Affecting operation

– Production by Risers – Gas-oil, oil/water separation – Normally, heave amplitude < 4m, pitch amplitude < 5 degrees,

roll amplitude < 10 degrees, excursion < (5~8)% water depth • Vibration of superstructures • Fatigue life of hull structures, risers, etc. • Survival in extreme seas • Local extreme structure damage (bottom slamming,

breaking wave impact, green water on deck etc.) • Human safety

Hydrodynamic Forces on a Body in Unbounded Fluid

Ub (1) Uf =0, Ub(t) = 06

Uf

F F (t) = −ma dUb (t) dt

Added massma : Depending on bodyD geometry, motion

direction, fluid density (2) Uf (t) = 0, Ub=06

Morrison’s formula: dUf (t) dUf (t) ρ: Fluid density

F (t) = ρ∇ + ma ∇: Body volumedt dt Froude-Krylov force Added mass effect

(3) Uf (t) = 0, Ub(t) = 06 6 n o F (t) = ρ∇ dUd

f

t (t) + ma

dUdf

t (t) − dUd

b

t (t)

Potential Flow

� In typical marine engineering applications such as ships, offshore platforms,

ULRe = ν = 106∼10

Thus, viscous effect can be neglected in general.

� Flow can be considered as a irrotational flow (i.e. vorticity ∇ × ~v = 0) except under some special conditions where flow separation occurs.

� Fluid motion in the ocean is normally assumed as a potential flow:

Velocity: ~v(x, y, z, t) = ∇φ(x, y, z, t)

Continuity equation: ∇2φ = 0

Momentum eqaution: p(x,y,z,t) ∂φ 1 2 − gzρ = − ∂t − 2 |∇φ|

� The key is to solve the Laplace equation with certain boundary conditions for the velocity potential φ(x, y, z, t)

Linearized (Airy) Wave Theory

•Boundary-Value Problem (BVP) for linearized (Airy) wave: y=0

y=-h

•Given velocity potential φ, find free-surface elevation η and pressure p:

Solution of 2D Periodic Progressive (Airy) Waves

Potential:

Free-surface elevation: η = A cos(kx − ωt)

A = H/2: wave amplitude; k= 2π/λ: wavenumber; ω =2π/T : frequency

Dispersion relation: ω2 = gk tanh kh

Phase velocity: Vp≡ λ ω= kT =p

gk tanh kh

Characteristics of a Linear Plane Progressive Wave

•Velocity Field:

•Pressure Field:

Wave Energy

•Wave energy propagation speed: group velocity: Vg dω= dk

Example: Wave Loads on Vertical Wall

A vertical wall is located at x=0 in in a water of depth h:

η(x, t) = A cos(kx − ωt) + A cos(kx + ωt)

φ(x, y, t) = gA cosh k(y+h) [sin(kx − ωt) − sin(kx + ωt)]ω cosh kh

p(x, y, t) = −ρ ∂φ cosh k(y+h) [cos(kx − ωt) + cos(kx + ωt)]−ρgy∂t −ρgy = ρgA cosh kh

p(x = 0, y, t) = 2ρgA cos ωt cosh k(y+h) − ρgycosh kh R 0Fx = p(x = 0, y, t)dy−h R 0 R 0

= 2ρgA cos ωt −h cosh k(y + h)dy − −h ρgydycosh kh

2ρgA ρgh2

= tanh kh cos ωt +k 2

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2.019 Design of Ocean Systems Spring 2011

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